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Regime-switching processes contain two components: continuous component anddiscrete component, which can be used to describe a continuous dynamical systemin a random environment. Such processes have many different properties thangeneral diffusion processes, and much more difficulties are needed to beovercome due to the intensive interaction between continuous and discretecomponent. We give conditions for the existence and uniqueness of invariantmeasures for state-dependent regime-switching diffusion processes byconstructing a new Markov chain to control the evolution of the state-dependentswitching process. We also establish the strong convergence in the $L^1$-normof the Euler-Maruyama's approximation and estimate the order of error. Arefined application of Skorokhod's representation of jumping processes plays asubstantial role in this work.

Invariant measures and Euler-Maruyama's approximations of state-dependent regime-switching diffusions